Filters in Analysis and Topology
... an ultrafilter is called free. It is worth noting that any ultrafilter which is not free is generated by a singleton, as if there exists an a with X \ {a} 6∈ F then {a} ∈ F . The complements of singletons generate the cofinite filter, so if F is not generated by a singleton then it contains the cofi ...
... an ultrafilter is called free. It is worth noting that any ultrafilter which is not free is generated by a singleton, as if there exists an a with X \ {a} 6∈ F then {a} ∈ F . The complements of singletons generate the cofinite filter, so if F is not generated by a singleton then it contains the cofi ...
Weakly b-Continuous Functions - International Journal of Science
... sets. He obtained some characterizations of weakly BRcontinuous functions and established relationships among such functions and several other existing functions. In a similar manner here our purpose is to introduce and study generalizations in form of new classes of functions namely weakly b-contin ...
... sets. He obtained some characterizations of weakly BRcontinuous functions and established relationships among such functions and several other existing functions. In a similar manner here our purpose is to introduce and study generalizations in form of new classes of functions namely weakly b-contin ...
On countable dense and strong n
... Let a : G × X → X be a continuous action of a topological group on a space X. For every g ∈ G, the function x 7→ a(g, x) is a homeomorphism of X. We use gx as an abbreviation for a(g, x). The action is called transitive if for all x, y ∈ X there exists g ∈ G such that gx = y. Hence a space X on whic ...
... Let a : G × X → X be a continuous action of a topological group on a space X. For every g ∈ G, the function x 7→ a(g, x) is a homeomorphism of X. We use gx as an abbreviation for a(g, x). The action is called transitive if for all x, y ∈ X there exists g ∈ G such that gx = y. Hence a space X on whic ...
On Normal Stratified Pseudomanifolds
... For a detailed treatment of the results contained in this section, see [8]. Manifolds considered in this paper will always be topological manifolds. A topological space is stratified if it can be written as a disjoint union of manifolds which are related by an incidence condition. Definition 1.1. Le ...
... For a detailed treatment of the results contained in this section, see [8]. Manifolds considered in this paper will always be topological manifolds. A topological space is stratified if it can be written as a disjoint union of manifolds which are related by an incidence condition. Definition 1.1. Le ...
![PFA(S)[S] and Locally Compact Normal Spaces](http://s1.studyres.com/store/data/001741952_1-aeadbdb88f4d36d94e029d346a059e2c-300x300.png)
PFA(S)[S] and Locally Compact Normal Spaces
... a) X is the union of countably many ω-bounded subspaces, or b) X does not have countable extent, or c) X has a separable closed subspace which is not Lindelöf. Recall a space is ω-bounded if every countable subspace has compact closure. ω-bounded spaces are obviously countably compact. From [23] we ...
... a) X is the union of countably many ω-bounded subspaces, or b) X does not have countable extent, or c) X has a separable closed subspace which is not Lindelöf. Recall a space is ω-bounded if every countable subspace has compact closure. ω-bounded spaces are obviously countably compact. From [23] we ...
Automorphism groups of metric structures
... This is equivalent to saying that there exists an open set O such that A∆O is meagre (and that is how we will use it). Thus, if A is Baire-measurable and not meagre, there exists a nonempty open O ⊆ X such that A is comeagre in O, i.e. O \ A is meagre. Just like measurable sets, Baire-measurable set ...
... This is equivalent to saying that there exists an open set O such that A∆O is meagre (and that is how we will use it). Thus, if A is Baire-measurable and not meagre, there exists a nonempty open O ⊆ X such that A is comeagre in O, i.e. O \ A is meagre. Just like measurable sets, Baire-measurable set ...
Splitting of the Identity Component in Locally Compact Abelian Groups
... x QR/ZY' for some n&No and cardinal number m, and likewise these are exactly the connected LCA groups splitting in every LCA group in which they are contained as identity components (compare Ahem-Jewitt [1], Dixmier[2] and Fulp-GrifSth [4]). It is weU known that this statement remains true if «L(3A ...
... x QR/ZY' for some n&No and cardinal number m, and likewise these are exactly the connected LCA groups splitting in every LCA group in which they are contained as identity components (compare Ahem-Jewitt [1], Dixmier[2] and Fulp-GrifSth [4]). It is weU known that this statement remains true if «L(3A ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.